Let $V$ be a vector space over a field $k$, and let $T:V\rightarrow V$ be linear, and let $f\in k[x]$. Suppose that $\lambda\in k$ is an eigenvalue of $T$ and let $v\in V$ be a corresponding eigenvector. Show that $f(T)v=f(\lambda)v$.
I'm not sure where to start. All I know is that $Tv=\lambda v$. I was thinking it might be easier to show that $f(T-\lambda I)=0$, but I still don't know where to start. Any help is appreciated!